Heres a non-technical summary of the basic idea of how to resolve
issue 5.3 along the lines currently being worked out. I am sending
this now in case the issue comes up at the telecon.
1. The OWL 'universe' consists of some subclasses of the RDFS
universe; the OWL vocabulary is restricted (by domain and range
assertions, mostly) to this sub-universe (consisting of owl:Thing,
owl:Class, owl:Property). However, exactly how the OWL universe
relates to the RDFS universe is not yet specified.
2. The 'weak' meanings of the OWL vocabulary can be given without
making any assumptions about the relationship between the OWL
universe and the RDFS universe. These meanings correspond roughly to
the DAML-style semantics that Peter refers to in issue 5.10, ie they
do not incorporate any assumptions about the existence of classes
that are not explicitly named. So they do not support many of the
'intuitive' entailments that Peter has pointed out. (Though see 7.
below.)
3. These weaker meanings can be applied to the entire RDFS domain if
one wishes, ie it is perfectly coherent to assert things like
rdfs:Class rdfs:subClassOf owl:Class .
in order to claim that the OWL universe and the RDFS universe are the
same. This allows the OWL vocabulary to be used (with its weak
meaning) in situations where classes may be instances, for example.
3a. The 'Russell paradox' RDF graph is a simple contradiction in this
semantics.
4. If one makes additional assumptions about the OWL universe, so
that its various parts are disjoint in the ways described in the OWL
MT, and that this restricted universe is closed under the various
class constructors (such as union/intersection and restrictions) then
one gets the 'strong' meanings of full OWL, but on a more restricted
universe. This supports a lot more entailments and also allows
efficient DL reasoners to work properly, providing a computational
guarantee of inferential power, but only on a suitably restricted
('layered') use of the OWL vocabulary. Still, many users may be happy
to stay inside the restrictions in order to gain the extra utility.
4a. If you were to add these assumptions to the unrestricted RDFS
universe, the Russell paradox graph would be a genuine paradox, since
these semantic assumptions would require it to be true.
5. There is a syntactic criterion which can be applied to an OWL/RDF
graph to detect when it 'fits' inside the restricted full-OWL
category (and therefore it is kosher to use the stronger inference
procedures on it) but the one I have is rather complicated and may be
too expensive to use in practice. Probably a simpler criterion could
be invented; I think Peter has one, but I can't follow the details.
Of course, one way to say this would be to refer to the translation
from the OWL abstract syntax, but it would be nice to have a quick
check that could be applied directly to an RDF graph.
6. It is possible to transfer more of the 'strong' meaning to the
unrestricted RDFS-wide use of the OWL vocabulary - for example, it is
certainly harmless to add closure under unions and intersections -
but determining the exact limits of this is a research problem.
Basically, we have to invent a new formal set theory and prove it
consistent. (My own view is that this isn't worth doing: if we need
to get this technical in order to provide a web language for general
use, we are barking up the wrong tree.)
7. I am pretty confident that (modulo minor bugs) this semantics
agrees with the picture one would get by starting from the abstract
syntax, applying Peter's MT to it, and then translating it into RDF.
That is, it produces the same entailments on any OWL/RDF that is a
translation of any well-formed abstract OWL. I confess however that I
have not written out a detailed proof of this.
8. When all this is done by translation into Lbase (or equivalent),
the 'weak' meanings correspond to straightforward FOL transcriptions
of the semantic meanings, and the 'strong' meanings involve adding a
number of extra comprehension axioms, most of them of the form
(forall <class, property> (exists <class> .....))
9. There are some unresolved issues about the need to 'strengthen'
parts of the RDFS MT. We are working on those. They are not
show-stoppers but we need to get the details right.
Write-up coming soon (Friday).
Pat
PS. Peter's help has gone from the acknowledgments stage to the
co-author stage at this point, but he isn't responsible for *this*
message and may want to disown parts of it.
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